In this paper, the wave structures of one-dimensional elastic–perfectly plastic solid Riemann problem are analyzed in elastic, plastic and elastic–plastic phases respectively. The research effort focuses on basic properties of the shock transition for the Wilkins model with Mie–Grüneisen equation of state (EOS). As a result of shock theory, the characters of Hugoniot curve and function of pure elastic and plastic are discussed. Especially, the wave structure in the presence of very strong shock transition from the elastic state to plastic state is researched, and the corresponding Hugoniot curve is given. Based on these analyses, the complete exact Riemann solution for the model is obtained, in which forty-nine possible solution types in total are found and enumerated for the first time. In addition, we exhibit the criteria for each wave-type and construct the complete exact Riemann solver. Several numerical tests are presented to prove the correctness of this exact Riemann solver.

本文分别从弹性、塑性和弹塑性相分析了一维弹-完全塑性固体黎曼问题的波结构。研究工作侧重于具有 Mie-Grüneisen 状态方程 (EOS) 的 Wilkins 模型的激波转变的基本属性。结合激波理论，讨论了Hugoniot曲线的性质和纯弹塑性函数。尤其是研究了存在极强冲击时由弹性态向塑性态转变的波浪结构，并给出了相应的Hugoniot曲线。在这些分析的基础上，得到了该模型的完整精确黎曼解，其中首次发现并列举了总共 49 种可能的解类型。此外，我们展示了每种波类型的标准并构建了完整的精确黎曼求解器。提供了几个数值试验来证明这个精确的黎曼求解器的正确性。